Low-resolution description of the conformational space for intrinsically disordered proteins.

Intrinsically disordered proteins (IDP) are at the center of numerous biological processes, and attract consequently extreme interest in structural biology. Numerous approaches have been developed for generating sets of IDP conformations verifying a given set of experimental measurements. We propose here to perform a systematic enumeration of protein conformations, carried out using the TAiBP approach based on distance geometry. This enumeration was performed on two proteins, Sic1 and pSic1, corresponding to unphosphorylated and phosphorylated states of an IDP. The relative populations of the obtained conformations were then obtained by fitting SAXS curves as well as Ramachandran probability maps, the original finite mixture approach RamaMix being developed for this second task. The similarity between profiles of local gyration radii provides to a certain extent a converged view of the Sic1 and pSic1 conformational space. Profiles and populations are thus proposed for describing IDP conformations. Different variations of the resulting gyration radius between phosphorylated and unphosphorylated states are observed, depending on the set of enumerated conformations as well as on the methods used for obtaining the populations.

Secondary structure assignment of proteins in the absence of sequence information.

Motivation: The structure of proteins is organized in a hierarchy among which the secondary structure elements, α-helix, ß-strand and loop, are the basic bricks. The determination of secondary structure elements usually requires the knowledge of the whole structure. Nevertheless, in numerous experimental circumstances, the protein structure is partially known. The detection of secondary structures from these partial structures is hampered by the lack of information about connecting residues along the primary sequence. Results: We introduce a new methodology to estimate the secondary structure elements from the values of local distances and angles between the protein atoms. Our method uses a message passing neural network, named Sequoia, which allows the automatic prediction of secondary structure elements from the values of local distances and angles between the protein atoms. This neural network takes as input the topology of the given protein graph, where the vertices are protein residues, and the edges are weighted by values of distances and pseudo-dihedral angles generalizing the backbone angles ϕ and ψ. Any pair of residues, independently of its covalent bonds along the primary sequence of the protein, is tagged with this distance and angle information. Sequoia permits the automatic detection of the secondary structure elements, with an F1-score larger than 80% for most of the cases, when α helices and ß strands are predicted. In contrast to the approaches classically used in structural biology, such as DSSP, Sequoia is able to capture the variations of geometry at the interface of adjacent secondary structure element. Due to its general modeling frame, Sequoia is able to handle graphs containing only C α atoms, which is particularly useful on low resolution structural input and in the frame of electron microscopy development. Availability and implementation: Sequoia source code can be found at https://github.com/Khalife/Sequoia with additional documentation. Supplementary information: Supplementary data are available at Bioinformatics Advances online.

Systematic Exploration of Protein Conformational Space Using a Distance Geometry Approach.

The optimization approaches classically used during the determination of protein structure encounter various difficulties, especially when the size of the conformational space is large. Indeed, in such a case, algorithmic convergence criteria are more difficult to set up. Moreover, the size of the search space makes it difficult to achieve a complete exploration. The interval branch-and-prune (iBP) approach, based on the reformulation of the distance geometry problem (DGP) provides a theoretical frame for the generation of protein conformations, by systematically sampling the conformational space. When an appropriate subset of interatomic distances is known exactly, this worst-case exponential-time algorithm is provably complete and fixed-parameter tractable. These guarantees, however, immediately disappear as distance measurement errors are introduced. Here we propose an improvement of this approach: threading-augmented interval branch-and-prune (TAiBP), where the combinatorial explosion of the original iBP approach arising from its exponential complexity is alleviated by partitioning the input instances into consecutive peptide fragments and by using self-organizing maps (SOMs) to obtain clusters of similar solutions. A validation of the TAiBP approach is presented here on a set of proteins of various sizes and structures. The calculation inputs are a uniform covalent geometry extracted from force field covalent terms, the backbone dihedral angles with error intervals, and a few long-range distances. For most of the proteins smaller than 50 residues and interval widths of 20°, the TAiBP approach yielded solutions with RMSD values smaller than 3 Å with respect to the initial protein conformation. The efficiency of the TAiBP approach for proteins larger than 50 residues will require the use of nonuniform covalent geometry and may have benefits from the recent development of residue-specific force-fields.

Minimal NMR distance information for rigidity of protein graphs.

Nuclear Magnetic Resonance (NMR) experiments provide distances between nearby atoms of a protein molecule. The corresponding structure determination problem is to determine the 3D protein structure by exploiting such distances. We present a new order on the atoms of the protein, based on information from the chemistry of proteins and NMR experiments, which allows us to formulate the problem as a combinatorial search. Additionally, this order tells us what kind of NMR distance information is crucial to understand the cardinality of the solution set of the problem and its computational complexity.

An algorithm to enumerate all possible protein conformations verifying a set of distance constraints.

BACKGROUND: The determination of protein structures satisfying distance constraints is an important problem in structural biology. Whereas the most common method currently employed is simulated annealing, there have been other methods previously proposed in the literature. Most of them, however, are designed to find one solution only. RESULTS: In order to explore exhaustively the feasible conformational space, we propose here an interval Branch-and-Prune algorithm (iBP) to solve the Distance Geometry Problem (DGP) associated to protein structure determination. This algorithm is based on a discretization of the problem obtained by recursively constructing a search space having the structure of a tree, and by verifying whether the generated atomic positions are feasible or not by making use of pruning devices. The pruning devices used here are directly related to features of protein conformations. CONCLUSIONS: We described the new algorithm iBP to generate protein conformations satisfying distance constraints, that would potentially allows a systematic exploration of the conformational space. The algorithm iBP has been applied on three α-helical peptides.

A discrete search algorithm for finding the structure of protein backbones and side chains.

Some information about protein structure can be obtained by using Nuclear Magnetic Resonance (NMR) techniques, but they provide only a sparse set of distances between atoms in a protein. The Molecular Distance Geometry Problem (MDGP) consists in determining the three-dimensional structure of a molecule using a set of known distances between some atoms. Recently, a Branch and Prune (BP) algorithm was proposed to calculate the backbone of a protein, based on a discrete formulation for the MDGP. We present an extension of the BP algorithm that can calculate not only the protein backbone, but the whole three-dimensional structure of proteins.

Algorithm for parametric community detection in networks.

Modularity maximization is extensively used to detect communities in complex networks. It has been shown, however, that this method suffers from a resolution limit: Small communities may be undetectable in the presence of larger ones even if they are very dense. To alleviate this defect, various modifications of the modularity function have been proposed as well as multiresolution methods. In this paper we systematically study a simple model (proposed by Pons and Latapy [Theor. Comput. Sci. 412, 892 (2011)] and similar to the parametric model of Reichardt and Bornholdt [Phys. Rev. E 74, 016110 (2006)]) with a single parameter α that balances the fraction of within community edges and the expected fraction of edges according to the configuration model. An exact algorithm is proposed to find optimal solutions for all values of α as well as the corresponding successive intervals of α values for which they are optimal. This algorithm relies upon a routine for exact modularity maximization and is limited to moderate size instances. An agglomerative hierarchical heuristic is therefore proposed to address parametric modularity detection in large networks. At each iteration the smallest value of α for which it is worthwhile to merge two communities of the current partition is found. Then merging is performed and the data are updated accordingly. An implementation is proposed with the same time and space complexity as the well-known Clauset-Newman-Moore (CNM) heuristic [Phys. Rev. E 70, 066111 (2004)]. Experimental results on artificial and real world problems show that (i) communities are detected by both exact and heuristic methods for all values of the parameter α; (ii) the dendrogram summarizing the results of the heuristic method provides a useful tool for substantive analysis, as illustrated particularly on a Les Misérables data set; (iii) the difference between the parametric modularity values given by the exact method and those given by the heuristic is moderate; (iv) the heuristic version of the proposed parametric method, viewed as a modularity maximization tool, gives better results than the CNM heuristic for large instances.

Exploiting symmetry properties of the discretizable molecular distance geometry problem.

The Discretizable Molecular Distance Geometry Problem (DMDGP) involves a subset of instances of the distance geometry problem for which some assumptions allowing for discretization are satisfied. The search domain for the DMDGP is a binary tree that can be efficiently explored by employing a Branch & Prune (BP) algorithm. We showed in recent works that this binary tree may contain several symmetries, which are directly related to the total number of solutions of DMDGP instances. In this paper, we study the possibility of exploiting these symmetries for speeding up the solution of DMDGPs, and propose an extension of the BP algorithm that we named symmetry-driven BP (symBP). Computational experiments on artificial and protein instances are presented.

Locally optimal heuristic for modularity maximization of networks.

Community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics, hybrids, and, in a few papers, exact algorithms. We consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way. This heuristic is compared with the spectral-based hierarchical divisive heuristic of Newman [Proc. Natl. Acad. Sci. USA 103, 8577 (2006).] and with the hierarchical agglomerative heuristic of Clauset, Newman, and Moore [Phys. Rev. E 70, 066111 (2004).]. Computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges. They show that the proposed divisive heuristic gives better results than the divisive heuristic of Newman and than the agglomerative heuristic of Clauset et al.

Loops and multiple edges in modularity maximization of networks.

The modularity maximization model proposed by Newman and Girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges. However, the applications usually correspond to simple graphs. These graphs are compared to a null model where the degree distribution is maintained but edges are placed at random. Therefore, in this null model there will be loops and possibly multiple edges. Sharp bounds on the expected number of loops, and their impact on the modularity, are derived. Then, building upon the work of Massen and Doye, but using algebra rather than simulation, we propose modified null models associated with graphs without loops but with multiple edges, graphs with loops but without multiple edges and graphs without loops nor multiple edges. We validate our models by using the exact algorithm for clique partitioning of Grötschel and Wakabayashi.

Edge ratio and community structure in networks.

A hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of community in the weak sense of Radicchi [Proc. Natl. Acad. Sci. U.S.A. 101, 2658 (2004)] is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. A mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition. This includes an exact solution of the problem of detecting indivisible communities. The resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets. For two problems of the former kind optimal solutions are found; for five problems of the latter kind the edge ratio algorithm always appears to be competitive. Moreover, it provides additional information in several cases, notably through the use of the dendrogram summarizing the resolution. Finally, both algorithms are compared on reduced versions of the data sets of Girvan and Newman [Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002)] and of Lancichinetti [Phys. Rev. E 78, 046110 (2008)]. Results for these instances appear to be comparable.

Column generation algorithms for exact modularity maximization in networks.

Finding modules, or clusters, in networks currently attracts much attention in several domains. The most studied criterion for doing so, due to Newman and Girvan [Phys. Rev. E 69, 026113 (2004)], is modularity maximization. Many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. There are few exact algorithms, prominent among which is a paper by Xu [Eur. Phys. J. B 60, 231 (2007)]. Modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of Grötschel and Wakabayashi [Math. Program. 45, 59 (1989)] applied. We propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming. Performance of the four resulting algorithms is compared on problems from the literature. Instances with up to 512 entities are solved exactly. Moreover, the computing time of previously solved problems are reduced substantially.